Transpose of a matrix is a new matrix rotating the rows of the given matrix.
Matrix
$$
A = \begin{bmatrix}
1 & 2 & 4 \\
8 & 2 & 1 \\
7 & 3 & 0 \\
2 & 1 & 2
\end{bmatrix}
$$
then transpose of this matrix is
$$
A^T =\begin{bmatrix}
1 & 8 & 7 & 2 \\
2 & 2 & 3 & 1 \\
4 & 1 & 0 & 2
\end{bmatrix}
$$
#Properties
$$
(A^T)^T = A
$$
$$
(A + B)^T = A^T + B^T
$$
$$
(A - B)^T = A^T - B^T
$$
$$
(kA)^T = kA^T
$$
$$
(AB)^T = B^TA^T
$$
$$
(A^T)^{-1} = (A^{-1})^T
$$
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